dist.Multivariate.t.Cholesky {LaplacesDemon}

Multivariate t Distribution: Cholesky Parameterization

Description

These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution, given the Cholesky parameterization.

Usage

dmvtc(x, mu, U, df=Inf, log=FALSE)
rmvtc(n=1, mu, U, df=Inf)

Arguments

x

This is either a vector of length k or a matrix with a number of columns, k, equal to the number of columns in scale matrix \textbf{S}.

n

This is the number of random draws.

mu

This is a numeric vector or matrix representing the location parameter,\mu (the mean vector), of the multivariate distribution (equal to the expected value when df > 1, otherwise represented as \nu > 1). When a vector, it must be of length k, or must have k columns as a matrix, as defined above.

U

This is the k \times k upper-triangular matrix that is Cholesky factor \textbf{U} of scale matrix \textbf{S}, such that S*df/(df-2) is the variance-covariance matrix when df > 2.

df

This is the degrees of freedom, and is often represented with \nu.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Multivariate

• Density:

  p(\theta) = \frac{\Gamma[(\nu+k)/2]}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}|\Sigma|^{1/2}[1 + (1/\nu)(\theta-\mu)^{\mathrm{T}} \Sigma^{-1} (\theta-\mu)]^{(\nu+k)/2}}

• Inventor: Unknown (to me, anyway)

• Notation 1: \theta \sim \mathrm{t}_k(\mu, \Sigma, \nu)

• Notation 2: p(\theta) = \mathrm{t}_k(\theta | \mu, \Sigma, \nu)

• Parameter 1: location vector \mu

• Parameter 2: positive-definite k \times k scale matrix \Sigma

• Parameter 3: degrees of freedom \nu > 0 (df in the functions)

• Mean: E(\theta) = \mu, for \nu > 1, otherwise undefined

• Variance: var(\theta) = \frac{\nu}{\nu - 2} \Sigma, for \nu > 2

• Mode: mode(\theta) = \mu

The multivariate t distribution, also called the multivariate Student or multivariate Student t distribution, is a multidimensional extension of the one-dimensional or univariate Student t distribution. A random vector is considered to be multivariate t-distributed if every linear combination of its components has a univariate Student t-distribution. This distribution has a mean parameter vector \mu of length k, and an upper-triangular k \times k matrix that is Cholesky factor \textbf{U}, as per the chol function for Cholesky decomposition. When degrees of freedom \nu=1, this is the multivariate Cauchy distribution.

In practice, \textbf{U} is fully unconstrained for proposals when its diagonal is log-transformed. The diagonal is exponentiated after a proposal and before other calculations. Overall, the Cholesky parameterization is faster than the traditional parameterization. Compared with dmvt, dmvtc must additionally matrix-multiply the Cholesky back to the scale matrix, but it does not have to check for or correct the scale matrix to positive-definiteness, which overall is slower. The same is true when comparing rmvt and rmvtc.

Value

dmvtc gives the density and rmvtc generates random deviates.

Author(s)

Statisticat, LLC. [e-mail]

See Also

chol, dinvwishartc, dmvc, dmvcp, dmvtp, dst, dstp, and dt.

Examples

library(LaplacesDemon)
x <- seq(-2,4,length=21)
y <- 2*x+10
z <- x+cos(y) 
mu <- c(1,12,2)
S <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3)
U <- chol(S)
df <- 4
f <- dmvtc(cbind(x,y,z), mu, U, df)
X <- rmvtc(1000, c(0,1,2), U, 5)
joint.density.plot(X[,1], X[,2], color=TRUE)